2.3. Resonances of a 2d dumbbell#
by M. Wess, 2024
This Notebook is part of the td_evp documentation on the implementation of time-domain methods for resonance problems in NGSolve.
Introduction#
This notebook is designed to present an implementation of the algorithm for solving resonance problems presented in
[NW24a] Lothar Nannen and Markus Wess. A krylov eigenvalue solver based on filtered time domain solutions. 2024. arXiv:2402.08515.
In particular, we solve the two dimensional problems from Section 3.1.
Implementation#
We start bei doing the necessary imports
from ngsolve import *
from ngsolve.webgui import Draw
from netgen.occ import *
import numpy as np
import scipy.linalg as spl
import matplotlib.pyplot as pl
import time as ti
%matplotlib notebook
from scipy.special import jn_zeros, jnjnp_zeros,jnp_zeros
The filtered operator#
class FilteredC(BaseMatrix):
def __init__(self,mata,matm_inv,tau,weights,freedofs=None):
super().__init__()
self.dt = tau
self.weights = weights
self.nsteps = len(weights)
self.mata = mata
self.matm_inv = matm_inv
self.freedofs=freedofs
self.vecu = self.mata.CreateColVector()
self.tmpvec1 = self.mata.CreateColVector()
self.tmpvec2 = self.mata.CreateColVector()
def CreateColVector(self):
return self.mata.CreateColVector()
def Shape(self):
return self.mata.shape
def CreateVector(self,col):
return self.mata.CreateVector(col)
def Mult(self,rhs,out):
with TaskManager():
self.vecu.data = rhs
tau = self.dt
out.data = tau*weights[0]*self.vecu
t = 0
unew = self.tmpvec1
uold = self.tmpvec2
uold.data = self.vecu
with TaskManager():
for i in range(1,self.nsteps):
t += tau
print("\r time = {}, step = {}".format(t,i),end="")
unew.data = 2*self.vecu - uold
unew.data -= tau**2 * self.matm_inv@self.mata * self.vecu
if self.freedofs:
unew.data[~self.freedofs] = 0.
uold.data = self.vecu
self.vecu.data = unew.data
out.data += tau*weights[i]*self.vecu
Geometry and mesh#
r_l = 1.5
r_r = 0.15
d = 0.03
w = 0.03
circ_left = Circle( (-r_l-w,0), r_l).Face()
circ_right = Circle( (r_r+w,0), r_r).Face()
handle = MoveTo(-r_l,-d).Rectangle(r_l+r_r,2*d).Face()
dumbbell = circ_left+handle+circ_right
geo = OCCGeometry(dumbbell,dim=2)
Draw(dumbbell);
maxh = 2*d
mesh = Mesh(geo.GenerateMesh(maxh = maxh))
mesh.Curve(2)
Draw(mesh);
order = 2
fes = H1LumpingFESpace(mesh,order=order)
print(fes.ndof)
u,v = fes.TnT()
mass = BilinearForm(u*v*dx(intrules=fes.GetIntegrationRules()),diagonal=True).Assemble().mat
stiffness = BilinearForm(grad(u)*grad(v)*dx).Assemble().mat
massinv = mass.Inverse()
14613
Determine time step by power iteration#
tol = 1e-4
tmpv = stiffness.CreateVector()
tmpv2 = stiffness.CreateVector()
tmpv.SetRandom()
tmpv /= tmpv.Norm()
mus = [1]
for i in range(500):
#print("i = {}".format(i))
tmpv2.data = tmpv
tmpv.data = massinv@stiffness*tmpv2
mu = InnerProduct(tmpv2,tmpv)/InnerProduct(tmpv2,tmpv2)
tmpv /= tmpv.Norm()
mus.append(mu)
if mus[-1]/mus[-2]<1+tol:
break
lammax = mus[-1]*(1+tol*200)
print("estimated largest eigenvalue: ",lammax)
tau = np.sqrt(4/lammax)
print("by power iteration stable for tau = {}".format(tau))
estimated largest eigenvalue: 49228.18216580044
by power iteration stable for tau = 0.009014115036549765
Define and plot filter function#
w_min = 0
w_max = 3
L = 200
endT = L*tau
weightf = lambda t: 4/np.pi*np.cos((w_max+w_min)/2*t)*(w_max-w_min)/2*np.sinc((w_max-w_min)/2*t/np.pi)
weights = weightf(tau*np.arange(L))
def beta(omega):
if np.isscalar(omega):
q = 1
else:
q = np.ones(omega.shape)
q_old = q
out = tau*weights[0]*q
for alpha in weights[1:]:
q_new = 2*q-tau**2*omega**2*q-q_old
q_old = q
q = q_new
out += tau*alpha*q
return out
omegas = np.arange(0,20,0.01)
betas = beta(omegas)
pl.plot(omegas,betas);
It remains to start the Krylov loop#
maxsteps = 25
C = FilteredC(stiffness,massinv,tau,weights)
now = ti.time()
errsmin = []
resmin = []
tol = 1e-6
solveevery = 1
tmpvec = mass.CreateVector()
vecs = MultiVector(tmpvec,0)
tmpvec.SetRandom()
tmpvec.data = 1/tmpvec.Norm()*tmpvec
vecs.Append(tmpvec)
pl.figure()
pl.plot(omegas,betas);
pl.xlim((0,10))
for i in range(maxsteps):
print("\n Krylowstep = {}".format(i))
tmpvec.data = C*vecs[-1]
vecs.AppendOrthogonalize(tmpvec)
if i%solveevery == 0:
tvecs = MultiVector(tmpvec,len(vecs))
tvecs.data = stiffness*vecs
tvecs2 = MultiVector(tmpvec,len(vecs))
tvecs2.data = mass*vecs
matm_proj = InnerProduct(tvecs2,vecs)
mats_proj = InnerProduct(tvecs,vecs)
lam,v = spl.eigh(mats_proj.NumPy(),matm_proj.NumPy())
lam = np.sqrt(np.abs(lam))
for li in lam:
pl.plot(li.real,1/maxsteps*i,'xr')
Krylowstep = 0
time = 0.009014115036549765, step = 1
time = 0.01802823007309953, step = 2
time = 0.027042345109649292, step = 3
time = 0.03605646014619906, step = 4
time = 0.045070575182748825, step = 5
time = 0.05408469021929859, step = 6
time = 0.06309880525584835, step = 7
time = 0.07211292029239812, step = 8
time = 0.08112703532894788, step = 9
time = 0.09014115036549765, step = 10
time = 0.09915526540204742, step = 11
time = 0.10816938043859718, step = 12
time = 0.11718349547514695, step = 13
time = 0.1261976105116967, step = 14
time = 0.13521172554824645, step = 15
time = 0.1442258405847962, step = 16
time = 0.15323995562134596, step = 17
time = 0.1622540706578957, step = 18
time = 0.17126818569444546, step = 19
time = 0.18028230073099522, step = 20
time = 0.18929641576754497, step = 21
time = 0.19831053080409472, step = 22
time = 0.20732464584064447, step = 23
time = 0.21633876087719422, step = 24
time = 0.22535287591374398, step = 25
time = 0.23436699095029373, step = 26
time = 0.24338110598684348, step = 27
time = 0.25239522102339323, step = 28
time = 0.261409336059943, step = 29
time = 0.27042345109649274, step = 30
time = 0.2794375661330425, step = 31
time = 0.28845168116959224, step = 32
time = 0.297465796206142, step = 33
time = 0.30647991124269175, step = 34
time = 0.3154940262792415, step = 35
time = 0.32450814131579125, step = 36
time = 0.333522256352341, step = 37
time = 0.34253637138889076, step = 38
time = 0.3515504864254405, step = 39
time = 0.36056460146199026, step = 40
time = 0.36957871649854, step = 41
time = 0.37859283153508977, step = 42
time = 0.3876069465716395, step = 43
time = 0.3966210616081893, step = 44
time = 0.405635176644739, step = 45
time = 0.4146492916812888, step = 46
time = 0.42366340671783853, step = 47
time = 0.4326775217543883, step = 48
time = 0.44169163679093804, step = 49
time = 0.4507057518274878, step = 50
time = 0.45971986686403754, step = 51
time = 0.4687339819005873, step = 52
time = 0.47774809693713705, step = 53
time = 0.4867622119736868, step = 54
time = 0.49577632701023655, step = 55
time = 0.5047904420467864, step = 56
time = 0.5138045570833362, step = 57
time = 0.522818672119886, step = 58
time = 0.5318327871564358, step = 59
time = 0.5408469021929856, step = 60
time = 0.5498610172295354, step = 61
time = 0.5588751322660852, step = 62
time = 0.567889247302635, step = 63
time = 0.5769033623391848, step = 64
time = 0.5859174773757346, step = 65
time = 0.5949315924122844, step = 66
time = 0.6039457074488342, step = 67
time = 0.612959822485384, step = 68
time = 0.6219739375219339, step = 69
time = 0.6309880525584837, step = 70
time = 0.6400021675950335, step = 71
time = 0.6490162826315833, step = 72
time = 0.6580303976681331, step = 73
time = 0.6670445127046829, step = 74
time = 0.6760586277412327, step = 75
time = 0.6850727427777825, step = 76
time = 0.6940868578143323, step = 77
time = 0.7031009728508821, step = 78
time = 0.7121150878874319, step = 79
time = 0.7211292029239817, step = 80
time = 0.7301433179605316, step = 81
time = 0.7391574329970814, step = 82
time = 0.7481715480336312, step = 83
time = 0.757185663070181, step = 84
time = 0.7661997781067308, step = 85
time = 0.7752138931432806, step = 86
time = 0.7842280081798304, step = 87
time = 0.7932421232163802, step = 88
time = 0.80225623825293, step = 89
time = 0.8112703532894798, step = 90
time = 0.8202844683260296, step = 91
time = 0.8292985833625794, step = 92
time = 0.8383126983991293, step = 93
time = 0.8473268134356791, step = 94
time = 0.8563409284722289, step = 95
time = 0.8653550435087787, step = 96
time = 0.8743691585453285, step = 97
time = 0.8833832735818783, step = 98
time = 0.8923973886184281, step = 99
time = 0.9014115036549779, step = 100
time = 0.9104256186915277, step = 101
time = 0.9194397337280775, step = 102
time = 0.9284538487646273, step = 103
time = 0.9374679638011771, step = 104
time = 0.946482078837727, step = 105
time = 0.9554961938742768, step = 106
time = 0.9645103089108266, step = 107
time = 0.9735244239473764, step = 108
time = 0.9825385389839262, step = 109
time = 0.991552654020476, step = 110
time = 1.0005667690570257, step = 111
time = 1.0095808840935754, step = 112
time = 1.018594999130125, step = 113
time = 1.0276091141666748, step = 114
time = 1.0366232292032245, step = 115
time = 1.0456373442397742, step = 116
time = 1.0546514592763239, step = 117
time = 1.0636655743128736, step = 118
time = 1.0726796893494233, step = 119
time = 1.081693804385973, step = 120
time = 1.0907079194225227, step = 121
time = 1.0997220344590724, step = 122
time = 1.108736149495622, step = 123
time = 1.1177502645321717, step = 124
time = 1.1267643795687214, step = 125
time = 1.1357784946052711, step = 126
time = 1.1447926096418208, step = 127
time = 1.1538067246783705, step = 128
time = 1.1628208397149202, step = 129
time = 1.17183495475147, step = 130
time = 1.1808490697880196, step = 131
time = 1.1898631848245693, step = 132
time = 1.198877299861119, step = 133
time = 1.2078914148976687, step = 134
time = 1.2169055299342184, step = 135
time = 1.225919644970768, step = 136
time = 1.2349337600073178, step = 137
time = 1.2439478750438675, step = 138
time = 1.2529619900804172, step = 139
time = 1.261976105116967, step = 140
time = 1.2709902201535166, step = 141
time = 1.2800043351900663, step = 142
time = 1.289018450226616, step = 143
time = 1.2980325652631657, step = 144
time = 1.3070466802997154, step = 145
time = 1.316060795336265, step = 146
time = 1.3250749103728148, step = 147
time = 1.3340890254093645, step = 148
time = 1.3431031404459142, step = 149
time = 1.3521172554824639, step = 150
time = 1.3611313705190136, step = 151
time = 1.3701454855555633, step = 152
time = 1.379159600592113, step = 153
time = 1.3881737156286627, step = 154
time = 1.3971878306652123, step = 155
time = 1.406201945701762, step = 156
time = 1.4152160607383117, step = 157
time = 1.4242301757748614, step = 158
time = 1.4332442908114111, step = 159
time = 1.4422584058479608, step = 160
time = 1.4512725208845105, step = 161
time = 1.4602866359210602, step = 162
time = 1.46930075095761, step = 163
time = 1.4783148659941596, step = 164
time = 1.4873289810307093, step = 165
time = 1.496343096067259, step = 166
time = 1.5053572111038087, step = 167
time = 1.5143713261403584, step = 168
time = 1.523385441176908, step = 169
time = 1.5323995562134578, step = 170
time = 1.5414136712500075, step = 171
time = 1.5504277862865572, step = 172
time = 1.559441901323107, step = 173
time = 1.5684560163596566, step = 174
time = 1.5774701313962063, step = 175
time = 1.586484246432756, step = 176
time = 1.5954983614693057, step = 177
time = 1.6045124765058554, step = 178
time = 1.613526591542405, step = 179
time = 1.6225407065789548, step = 180
time = 1.6315548216155045, step = 181
time = 1.6405689366520542, step = 182
time = 1.6495830516886039, step = 183
time = 1.6585971667251536, step = 184
time = 1.6676112817617033, step = 185
time = 1.676625396798253, step = 186
time = 1.6856395118348027, step = 187
time = 1.6946536268713523, step = 188
time = 1.703667741907902, step = 189
time = 1.7126818569444517, step = 190
time = 1.7216959719810014, step = 191
time = 1.7307100870175511, step = 192
time = 1.7397242020541008, step = 193
time = 1.7487383170906505, step = 194
time = 1.7577524321272002, step = 195
time = 1.76676654716375, step = 196
time = 1.7757806622002996, step = 197
time = 1.7847947772368493, step = 198
time = 1.793808892273399, step = 199
Krylowstep = 1
time = 0.009014115036549765, step = 1
time = 0.01802823007309953, step = 2
time = 0.027042345109649292, step = 3
time = 0.03605646014619906, step = 4
time = 0.045070575182748825, step = 5
time = 0.05408469021929859, step = 6
time = 0.06309880525584835, step = 7
time = 0.07211292029239812, step = 8
time = 0.08112703532894788, step = 9
time = 0.09014115036549765, step = 10
time = 0.09915526540204742, step = 11
time = 0.10816938043859718, step = 12
time = 0.11718349547514695, step = 13
time = 0.1261976105116967, step = 14
time = 0.13521172554824645, step = 15
time = 0.1442258405847962, step = 16
time = 0.15323995562134596, step = 17
time = 0.1622540706578957, step = 18
time = 0.17126818569444546, step = 19
time = 0.18028230073099522, step = 20
time = 0.18929641576754497, step = 21
time = 0.19831053080409472, step = 22
time = 0.20732464584064447, step = 23
time = 0.21633876087719422, step = 24
time = 0.22535287591374398, step = 25
time = 0.23436699095029373, step = 26
time = 0.24338110598684348, step = 27
time = 0.25239522102339323, step = 28
time = 0.261409336059943, step = 29
time = 0.27042345109649274, step = 30
time = 0.2794375661330425, step = 31
time = 0.28845168116959224, step = 32
time = 0.297465796206142, step = 33
time = 0.30647991124269175, step = 34
time = 0.3154940262792415, step = 35
time = 0.32450814131579125, step = 36
time = 0.333522256352341, step = 37
time = 0.34253637138889076, step = 38
time = 0.3515504864254405, step = 39
time = 0.36056460146199026, step = 40
time = 0.36957871649854, step = 41
time = 0.37859283153508977, step = 42
time = 0.3876069465716395, step = 43
time = 0.3966210616081893, step = 44
time = 0.405635176644739, step = 45
time = 0.4146492916812888, step = 46
time = 0.42366340671783853, step = 47
time = 0.4326775217543883, step = 48
time = 0.44169163679093804, step = 49
time = 0.4507057518274878, step = 50
time = 0.45971986686403754, step = 51
time = 0.4687339819005873, step = 52
time = 0.47774809693713705, step = 53
time = 0.4867622119736868, step = 54
time = 0.49577632701023655, step = 55
time = 0.5047904420467864, step = 56
time = 0.5138045570833362, step = 57
time = 0.522818672119886, step = 58
time = 0.5318327871564358, step = 59
time = 0.5408469021929856, step = 60
time = 0.5498610172295354, step = 61
time = 0.5588751322660852, step = 62
time = 0.567889247302635, step = 63
time = 0.5769033623391848, step = 64
time = 0.5859174773757346, step = 65
time = 0.5949315924122844, step = 66
time = 0.6039457074488342, step = 67
time = 0.612959822485384, step = 68
time = 0.6219739375219339, step = 69
time = 0.6309880525584837, step = 70
time = 0.6400021675950335, step = 71
time = 0.6490162826315833, step = 72
time = 0.6580303976681331, step = 73
time = 0.6670445127046829, step = 74
time = 0.6760586277412327, step = 75
time = 0.6850727427777825, step = 76
time = 0.6940868578143323, step = 77
time = 0.7031009728508821, step = 78
time = 0.7121150878874319, step = 79
time = 0.7211292029239817, step = 80
time = 0.7301433179605316, step = 81
time = 0.7391574329970814, step = 82
time = 0.7481715480336312, step = 83
time = 0.757185663070181, step = 84
time = 0.7661997781067308, step = 85
time = 0.7752138931432806, step = 86
time = 0.7842280081798304, step = 87
time = 0.7932421232163802, step = 88
time = 0.80225623825293, step = 89
time = 0.8112703532894798, step = 90
time = 0.8202844683260296, step = 91
time = 0.8292985833625794, step = 92
time = 0.8383126983991293, step = 93
time = 0.8473268134356791, step = 94
time = 0.8563409284722289, step = 95
time = 0.8653550435087787, step = 96
time = 0.8743691585453285, step = 97
time = 0.8833832735818783, step = 98
time = 0.8923973886184281, step = 99
time = 0.9014115036549779, step = 100
time = 0.9104256186915277, step = 101
time = 0.9194397337280775, step = 102
time = 0.9284538487646273, step = 103
time = 0.9374679638011771, step = 104
time = 0.946482078837727, step = 105
time = 0.9554961938742768, step = 106
time = 0.9645103089108266, step = 107
time = 0.9735244239473764, step = 108
time = 0.9825385389839262, step = 109
time = 0.991552654020476, step = 110
time = 1.0005667690570257, step = 111
time = 1.0095808840935754, step = 112
time = 1.018594999130125, step = 113
time = 1.0276091141666748, step = 114
time = 1.0366232292032245, step = 115
time = 1.0456373442397742, step = 116
time = 1.0546514592763239, step = 117
time = 1.0636655743128736, step = 118
time = 1.0726796893494233, step = 119
time = 1.081693804385973, step = 120
time = 1.0907079194225227, step = 121
time = 1.0997220344590724, step = 122
time = 1.108736149495622, step = 123
time = 1.1177502645321717, step = 124
time = 1.1267643795687214, step = 125
time = 1.1357784946052711, step = 126
time = 1.1447926096418208, step = 127
time = 1.1538067246783705, step = 128
time = 1.1628208397149202, step = 129
time = 1.17183495475147, step = 130
time = 1.1808490697880196, step = 131
time = 1.1898631848245693, step = 132
time = 1.198877299861119, step = 133
time = 1.2078914148976687, step = 134
time = 1.2169055299342184, step = 135
time = 1.225919644970768, step = 136
time = 1.2349337600073178, step = 137
time = 1.2439478750438675, step = 138
time = 1.2529619900804172, step = 139
time = 1.261976105116967, step = 140
time = 1.2709902201535166, step = 141
time = 1.2800043351900663, step = 142
time = 1.289018450226616, step = 143
time = 1.2980325652631657, step = 144
time = 1.3070466802997154, step = 145
time = 1.316060795336265, step = 146
time = 1.3250749103728148, step = 147
time = 1.3340890254093645, step = 148
time = 1.3431031404459142, step = 149
time = 1.3521172554824639, step = 150
time = 1.3611313705190136, step = 151
time = 1.3701454855555633, step = 152
time = 1.379159600592113, step = 153
time = 1.3881737156286627, step = 154
time = 1.3971878306652123, step = 155
time = 1.406201945701762, step = 156
time = 1.4152160607383117, step = 157
time = 1.4242301757748614, step = 158
time = 1.4332442908114111, step = 159
time = 1.4422584058479608, step = 160
time = 1.4512725208845105, step = 161
time = 1.4602866359210602, step = 162
time = 1.46930075095761, step = 163
time = 1.4783148659941596, step = 164
time = 1.4873289810307093, step = 165
time = 1.496343096067259, step = 166
time = 1.5053572111038087, step = 167
time = 1.5143713261403584, step = 168
time = 1.523385441176908, step = 169
time = 1.5323995562134578, step = 170
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Krylowstep = 2
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Krylowstep = 3
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Krylowstep = 4
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Krylowstep = 5
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Krylowstep = 6
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Krylowstep = 7
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Krylowstep = 8
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Krylowstep = 9
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Krylowstep = 10
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Krylowstep = 15
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time = 1.2529619900804172, step = 139
time = 1.261976105116967, step = 140
time = 1.2709902201535166, step = 141
time = 1.2800043351900663, step = 142
time = 1.289018450226616, step = 143
time = 1.2980325652631657, step = 144
time = 1.3070466802997154, step = 145
time = 1.316060795336265, step = 146
time = 1.3250749103728148, step = 147
time = 1.3340890254093645, step = 148
time = 1.3431031404459142, step = 149
time = 1.3521172554824639, step = 150
time = 1.3611313705190136, step = 151
time = 1.3701454855555633, step = 152
time = 1.379159600592113, step = 153
time = 1.3881737156286627, step = 154
time = 1.3971878306652123, step = 155
time = 1.406201945701762, step = 156
time = 1.4152160607383117, step = 157
time = 1.4242301757748614, step = 158
time = 1.4332442908114111, step = 159
time = 1.4422584058479608, step = 160
time = 1.4512725208845105, step = 161
time = 1.4602866359210602, step = 162
time = 1.46930075095761, step = 163
time = 1.4783148659941596, step = 164
time = 1.4873289810307093, step = 165
time = 1.496343096067259, step = 166
time = 1.5053572111038087, step = 167
time = 1.5143713261403584, step = 168
time = 1.523385441176908, step = 169
time = 1.5323995562134578, step = 170
time = 1.5414136712500075, step = 171
time = 1.5504277862865572, step = 172
time = 1.559441901323107, step = 173
time = 1.5684560163596566, step = 174
time = 1.5774701313962063, step = 175
time = 1.586484246432756, step = 176
time = 1.5954983614693057, step = 177
time = 1.6045124765058554, step = 178
time = 1.613526591542405, step = 179
time = 1.6225407065789548, step = 180
time = 1.6315548216155045, step = 181
time = 1.6405689366520542, step = 182
time = 1.6495830516886039, step = 183
time = 1.6585971667251536, step = 184
time = 1.6676112817617033, step = 185
time = 1.676625396798253, step = 186
time = 1.6856395118348027, step = 187
time = 1.6946536268713523, step = 188
time = 1.703667741907902, step = 189
time = 1.7126818569444517, step = 190
time = 1.7216959719810014, step = 191
time = 1.7307100870175511, step = 192
time = 1.7397242020541008, step = 193
time = 1.7487383170906505, step = 194
time = 1.7577524321272002, step = 195
time = 1.76676654716375, step = 196
time = 1.7757806622002996, step = 197
time = 1.7847947772368493, step = 198
time = 1.793808892273399, step = 199
Krylowstep = 24
time = 0.009014115036549765, step = 1
time = 0.01802823007309953, step = 2
time = 0.027042345109649292, step = 3
time = 0.03605646014619906, step = 4
time = 0.045070575182748825, step = 5
time = 0.05408469021929859, step = 6
time = 0.06309880525584835, step = 7
time = 0.07211292029239812, step = 8
time = 0.08112703532894788, step = 9
time = 0.09014115036549765, step = 10
time = 0.09915526540204742, step = 11
time = 0.10816938043859718, step = 12
time = 0.11718349547514695, step = 13
time = 0.1261976105116967, step = 14
time = 0.13521172554824645, step = 15
time = 0.1442258405847962, step = 16
time = 0.15323995562134596, step = 17
time = 0.1622540706578957, step = 18
time = 0.17126818569444546, step = 19
time = 0.18028230073099522, step = 20
time = 0.18929641576754497, step = 21
time = 0.19831053080409472, step = 22
time = 0.20732464584064447, step = 23
time = 0.21633876087719422, step = 24
time = 0.22535287591374398, step = 25
time = 0.23436699095029373, step = 26
time = 0.24338110598684348, step = 27
time = 0.25239522102339323, step = 28
time = 0.261409336059943, step = 29
time = 0.27042345109649274, step = 30
time = 0.2794375661330425, step = 31
time = 0.28845168116959224, step = 32
time = 0.297465796206142, step = 33
time = 0.30647991124269175, step = 34
time = 0.3154940262792415, step = 35
time = 0.32450814131579125, step = 36
time = 0.333522256352341, step = 37
time = 0.34253637138889076, step = 38
time = 0.3515504864254405, step = 39
time = 0.36056460146199026, step = 40
time = 0.36957871649854, step = 41
time = 0.37859283153508977, step = 42
time = 0.3876069465716395, step = 43
time = 0.3966210616081893, step = 44
time = 0.405635176644739, step = 45
time = 0.4146492916812888, step = 46
time = 0.42366340671783853, step = 47
time = 0.4326775217543883, step = 48
time = 0.44169163679093804, step = 49
time = 0.4507057518274878, step = 50
time = 0.45971986686403754, step = 51
time = 0.4687339819005873, step = 52
time = 0.47774809693713705, step = 53
time = 0.4867622119736868, step = 54
time = 0.49577632701023655, step = 55
time = 0.5047904420467864, step = 56
time = 0.5138045570833362, step = 57
time = 0.522818672119886, step = 58
time = 0.5318327871564358, step = 59
time = 0.5408469021929856, step = 60
time = 0.5498610172295354, step = 61
time = 0.5588751322660852, step = 62
time = 0.567889247302635, step = 63
time = 0.5769033623391848, step = 64
time = 0.5859174773757346, step = 65
time = 0.5949315924122844, step = 66
time = 0.6039457074488342, step = 67
time = 0.612959822485384, step = 68
time = 0.6219739375219339, step = 69
time = 0.6309880525584837, step = 70
time = 0.6400021675950335, step = 71
time = 0.6490162826315833, step = 72
time = 0.6580303976681331, step = 73
time = 0.6670445127046829, step = 74
time = 0.6760586277412327, step = 75
time = 0.6850727427777825, step = 76
time = 0.6940868578143323, step = 77
time = 0.7031009728508821, step = 78
time = 0.7121150878874319, step = 79
time = 0.7211292029239817, step = 80
time = 0.7301433179605316, step = 81
time = 0.7391574329970814, step = 82
time = 0.7481715480336312, step = 83
time = 0.757185663070181, step = 84
time = 0.7661997781067308, step = 85
time = 0.7752138931432806, step = 86
time = 0.7842280081798304, step = 87
time = 0.7932421232163802, step = 88
time = 0.80225623825293, step = 89
time = 0.8112703532894798, step = 90
time = 0.8202844683260296, step = 91
time = 0.8292985833625794, step = 92
time = 0.8383126983991293, step = 93
time = 0.8473268134356791, step = 94
time = 0.8563409284722289, step = 95
time = 0.8653550435087787, step = 96
time = 0.8743691585453285, step = 97
time = 0.8833832735818783, step = 98
time = 0.8923973886184281, step = 99
time = 0.9014115036549779, step = 100
time = 0.9104256186915277, step = 101
time = 0.9194397337280775, step = 102
time = 0.9284538487646273, step = 103
time = 0.9374679638011771, step = 104
time = 0.946482078837727, step = 105
time = 0.9554961938742768, step = 106
time = 0.9645103089108266, step = 107
time = 0.9735244239473764, step = 108
time = 0.9825385389839262, step = 109
time = 0.991552654020476, step = 110
time = 1.0005667690570257, step = 111
time = 1.0095808840935754, step = 112
time = 1.018594999130125, step = 113
time = 1.0276091141666748, step = 114
time = 1.0366232292032245, step = 115
time = 1.0456373442397742, step = 116
time = 1.0546514592763239, step = 117
time = 1.0636655743128736, step = 118
time = 1.0726796893494233, step = 119
time = 1.081693804385973, step = 120
time = 1.0907079194225227, step = 121
time = 1.0997220344590724, step = 122
time = 1.108736149495622, step = 123
time = 1.1177502645321717, step = 124
time = 1.1267643795687214, step = 125
time = 1.1357784946052711, step = 126
time = 1.1447926096418208, step = 127
time = 1.1538067246783705, step = 128
time = 1.1628208397149202, step = 129
time = 1.17183495475147, step = 130
time = 1.1808490697880196, step = 131
time = 1.1898631848245693, step = 132
time = 1.198877299861119, step = 133
time = 1.2078914148976687, step = 134
time = 1.2169055299342184, step = 135
time = 1.225919644970768, step = 136
time = 1.2349337600073178, step = 137
time = 1.2439478750438675, step = 138
time = 1.2529619900804172, step = 139
time = 1.261976105116967, step = 140
time = 1.2709902201535166, step = 141
time = 1.2800043351900663, step = 142
time = 1.289018450226616, step = 143
time = 1.2980325652631657, step = 144
time = 1.3070466802997154, step = 145
time = 1.316060795336265, step = 146
time = 1.3250749103728148, step = 147
time = 1.3340890254093645, step = 148
time = 1.3431031404459142, step = 149
time = 1.3521172554824639, step = 150
time = 1.3611313705190136, step = 151
time = 1.3701454855555633, step = 152
time = 1.379159600592113, step = 153
time = 1.3881737156286627, step = 154
time = 1.3971878306652123, step = 155
time = 1.406201945701762, step = 156
time = 1.4152160607383117, step = 157
time = 1.4242301757748614, step = 158
time = 1.4332442908114111, step = 159
time = 1.4422584058479608, step = 160
time = 1.4512725208845105, step = 161
time = 1.4602866359210602, step = 162
time = 1.46930075095761, step = 163
time = 1.4783148659941596, step = 164
time = 1.4873289810307093, step = 165
time = 1.496343096067259, step = 166
time = 1.5053572111038087, step = 167
time = 1.5143713261403584, step = 168
time = 1.523385441176908, step = 169
time = 1.5323995562134578, step = 170
time = 1.5414136712500075, step = 171
time = 1.5504277862865572, step = 172
time = 1.559441901323107, step = 173
time = 1.5684560163596566, step = 174
time = 1.5774701313962063, step = 175
time = 1.586484246432756, step = 176
time = 1.5954983614693057, step = 177
time = 1.6045124765058554, step = 178
time = 1.613526591542405, step = 179
time = 1.6225407065789548, step = 180
time = 1.6315548216155045, step = 181
time = 1.6405689366520542, step = 182
time = 1.6495830516886039, step = 183
time = 1.6585971667251536, step = 184
time = 1.6676112817617033, step = 185
time = 1.676625396798253, step = 186
time = 1.6856395118348027, step = 187
time = 1.6946536268713523, step = 188
time = 1.703667741907902, step = 189
time = 1.7126818569444517, step = 190
time = 1.7216959719810014, step = 191
time = 1.7307100870175511, step = 192
time = 1.7397242020541008, step = 193
time = 1.7487383170906505, step = 194
time = 1.7577524321272002, step = 195
time = 1.76676654716375, step = 196
time = 1.7757806622002996, step = 197
time = 1.7847947772368493, step = 198
time = 1.793808892273399, step = 199
eigf = (vecs*Matrix(v.real)).Evaluate()
gfu = GridFunction(fes)
for i in range(1,5):
gfu.vec.data = eigf[i]
Draw(gfu)